1.6: Analyzing the Flows and Reactions in a Plant - Page 1

1.6.1 Summary of Degree of Freedom Table

Before we look at more complicated problems, we need to have a
systematic procedure for analyzing interconnected units. We need
particularly to have some way to determine which units in a plant can
be (and should be) analyzed first. Then we want to establish an order
for further analysis. The modules and units considered in the last
sections could all be analyzed completely. In many systems, we will
find that only a few units can be treated in that way. A system
(module, unit or plant) that can be analyzed completely to determine
all pertinent flow rates and reaction rates is said to be completely
determined. Such a system is also said to have a Degree of Freedom of
0.

The degree of freedom for a unit is a measure of how many
specifications we can (or need to) make so that all unknown flows and
reaction rates can be found. The larger the degree of freedom for a
unit, the less we know about it. Since flow and reaction rates are
the unknowns while balances, given compositions, flows, flow ratios
and conversions and splitter restrictions all reduce the uncertainty,
the differences are what we call the degree of freedom.

Just as each unit in a plant has unknowns (flow and reaction rates)
that we may try to solve for, the overall plant has a similar set of
unknowns. In fact all we need to do is look strictly at those flow
streams that enter and leave the plant and forget the streams that
lead from one unit to another. We may have to lump reaction rates if
the same reaction occurs in more than one unit because the overall
system does not care where a reaction takes place. We also can find
much information that is useless in our analysis of the overall
system. Only if the information specifies something about the plant
feed and product streams can it be used to reduce the degree of
freedom for the overall system.

An entirely different problem is connected with analyzing the
"process". In this case we look at all the unknowns in the plant.
This includes the flows of all compounds in all streams and the rates
of all reactions in all units. This time the same reaction is counted
as many times as it occurs. In addition all information is
usable.

The degree of freedom for the overall system and for the process are
difficult to find unless you are careful about the meaning of each
term in them. The following table shows a precise way to find the
degree of freedom for each unit, the overall system and the
process.

Specific definitions of the entries in the table are given next.
Some of these are obvious, others can be quite confusing unless you
are careful to think precisely about what they represent.

Flow Variables

FVk=

the sum of the number of compounds that are involved in
the balance equations in all the streams entering or leaving
unit k.

FVO=

the sum of the number of compounds that are involved in
the balance equations in all the streams entering or leaving
the overall system.

FVP=

the sum of the number of compounds that are involved in
the balance equations in all the streams anywhere in the
system including those entering or leaving the overall
system as well as those connecting the units.

Reaction Variables

Rk=

the number of independent reactions occurring inside unit
k.

RO=

the number of independent reactions occurring inside the
overall system. Do not count each reaction more than once
even if it occurs in more than one unit.

RP=

sum (Rk)

Balances

Bk=

the number of compounds that are found in the flow
streams in or out of unit k or take part in the reactions in
the unit.

Bk=

the number of compounds that are found in the flow
streams in or out of the overall system or take part in the
reactions inside the system.

BP=

sum(Bk)

Specified Compositions

CPk=

the number of compositions given for any flow stream into
or out of unit k. The maximum number that is counted in any
one stream is one less than the number of compounds in that
stream.

CPO=

the number of compositions given for any flow stream into
or out of the overall system.

CPP=

the number of compositions given for any flow stream
anywhere in the system.

Specified Flow Rates

Fk=

the number of flow rates given for the streams in or out
of unit k.

FO=

the number of flow rates given for the streams in or out
of the overall system.

FP=

the number of flow rates given for all the streams
anywhere in the system.

Specified Flow Ratios

FRk=

the number of flow ratios of streams into or out of unit
k. Note that both streams in the ratio must be involved with
unit k to be counted.

FRO=

the number of flow ratios of streams into or out of the
overall system. Note that both streams in the ratio must be
ones that enter or leave the overall system to be
counted.

FRP=

the number of flow ratios given for streams anywhere in
the system.

Specified Conversions

CVk=

the number of conversions specified for reactants fed to
unit k. Do not count a conversion of 100% if the reactant
that is the basis of the conversion does not appear in any
exit stream from the unit.

CVO=

the number of conversions specified for reactants fed to
the overall system. Do not count a conversion of 100% if the
reactant that is the basis of the conversion does not appear
in any exit stream from the system.

CVP=

sum(CVk)

Splitter Restrictions

Sk=

0 unless the unit is a stream splitter. If it is a
splitter, then it is (number streams
leaving - 1)(number of compounds in the stream
-1).

SP=

sum(Sk)

Degree of Freedom

°Fk = FVk +
Rk - (Bk + CPk + Fk +
FRk + CVk + Sk)

°FO = FVO + RO - (BO
+ CPO + FO + FRO + CVO +
SO)

°FP = FP + RP - (BP
+ CPP + FP + FRP + CVP +
SP)

Degree of Freedom Table

Unit Name:

Unit 1

Unit 2

Unit 3

Unit 4

Unit 5

Unit 6

Over-all

Process

Flow
Variables

Reaction
Variables

Balances

Given
Compositions

Given Flows

Given Flow
Ratios

Given
Conversions

Splitter
Restrictions

Net °F

This is a typical degree of freedom table for a system with no more
than 6 units.

1.6.2 Using the Modules to Simulate a Plant

Three procedures will be demonstrated in this section of the notes
for simulating the steady state behavior of an entire plant. Each
procedure uses the module functions shown in the previous section of
the notes. The three methods are:

Interactive Use of the Modules

Using a General Purpose Driver

Writing a Program for a Specific Plant

Each method has its advantages and disadvantages. The modules may
be readily combined to simulate almost any steady state chemical
system. As soon as we encounter systems with a large number of units,
we find that it is tedious to keep track of the order of the units
and the information that needs to be given as we proceed through the
system. It also becomes tedious if we find that a trial and error
solution must be used to find some of the flow or reaction variables.
Thus we need to adopt one of the last two procedures to be efficient.
If we have only a few systems to analyze, a specific program written
to simulate each system may be the best approach. This becomes less
practical as the number of systems to be analyzed grows. Then we will
find general purpose drivers will be very useful.

Two general purpose programs will be demonstrated in this
chapter:

Function

Used For

1

drive

Driver function for combining modules. (in
~ceng301/matlab/massmods)

2

vweg2

Vector root finding module. (in ~ceng301/matlab/misc)

Interactive Use of the Modules

Multi-unit systems can be simulated by combining several of the
modular programs. To illustrate this, let's do Example Problem
3.13 in the text. Let the streams be numbered as in Figure
3.7 in the text. The chlorination reactor has two inlet streams
and two outlet streams, but react can handle only one inlet and one
outlet stream. Therefore, streams 1 and 2 must be mixed; then this
mixture can be reacted. Afterwards the outlet from react can be
separated using sep.

Using start301 to set the compounds and reactions,
select New Session, followed by Mass Balances Only. We
then wish to use the CENG 301 database since all the compounds
we are going to use can be found in the database.

>> start301
Copyright 1999 Rice University
All rights reserved
Welcome to CENG301's start301!!
You have three choices, Please read them carefully
Then click on the appropriate choice in the menu bar
1: Click (1) to start a new session
Input the name of your new file: chloros
The output file name is: chloros
Input the number of compounds: 7
The number of compounds is: 7
Enter the name of compound # 1: C6H6
Enter the name of compound # 2: C6H5Cl
Enter the name of compound # 3: C6H4Cl2
Enter the name of compound # 4: C6H3Cl3
Enter the name of compound # 5: C6H2Cl4
Enter the name of compound # 6: Cl2
Enter the name of compound # 7: HCl
Enter the number of reactions: 4
Enter the coefficients for each compound in the same order that
the compounds are listed. Coefficients for reactants should be
Negative, and coefficients for products should be positive
Enter the coefficients for each compound in reaction # 1
C6H6 C6H5Cl C6H4Cl2 C6H3Cl3 C6H2Cl4 Cl2 HCl
-1 1 0 0 0 -1 1
Enter the coefficients for each compound in the same order that
the compounds are listed. Coefficients for reactants should be
Negative, and coefficients for products should be positive
Enter the coefficients for each compound in reaction # 2
C6H6 C6H5Cl C6H4Cl2 C6H3Cl3 C6H2Cl4 Cl2 HCl
0 -1 1 0 0 -1 1
Enter the coefficients for each compound in the same order that
the compounds are listed. Coefficients for reactants should be
Negative, and coefficients for products should be positive
Enter the coefficients for each compound in reaction # 3
C6H6 C6H5Cl C6H4Cl2 C6H3Cl3 C6H2Cl4 Cl2 HCl
0 0 -1 1 0 -1 1
Enter the coefficients for each compound in the same order that
the compounds are listed. Coefficients for reactants should be
Negative, and coefficients for products should be positive
Enter the coefficients for each compound in reaction # 4
C6H6 C6H5Cl C6H4Cl2 C6H3Cl3 C6H2Cl4 Cl2 HCl
0 0 0 -1 1 -1 1
Do you want to check to see if the coefficients are correct?
type "y" for yes or simply press enter to move on:
Your reactions are currently as follows:
1) C6H6 + Cl2 --> C6H5Cl + HCl
2) C6H5Cl + Cl2 --> C6H4Cl2 + HCl
3) C6H4Cl2 + Cl2 --> C6H3Cl3 + HCl
4) C6H3Cl3 + Cl2 --> C6H2Cl4 + HCl
Type "y" to change an equation, or press "enter" to continue:
Here are your compounds' formulae and names:
No. Formula Name
----------------------------------------
1 C6H6 benzene
2 C6H5Cl chlorobenzene
3 C6H4Cl2 m-dichlorobenzene
4 C6H3Cl3 trichlorobenzene
5 C6H2Cl4 tetrachlorobenzene
6 Cl2 chlorine
7 HCl hydrogen chloride
Here are your reactions:
----------------------------------------
1) C6H6 + Cl2 --> C6H5Cl + HCl
2) C6H5Cl + Cl2 --> C6H4Cl2 + HCl
3) C6H4Cl2 + Cl2 --> C6H3Cl3 + HCl
4) C6H3Cl3 + Cl2 --> C6H2Cl4 + HCl
Enter the number of streams: 6
The variables for your compounds have now been created,
you may continue, or come back later and reload the same data.

The flow of Benzene in stream 1 is 1000 and of Chlorine in stream
2 is 3600. These may be set by: